I don't know all the factorial shortcuts. Since I'm using a fixed setup, I'm not getting too complex. If I wanted a list of combinations, I'd write a program to generate one. If I wanted random, I could sort the column based on random numbers.It's a pity you resigned from the 2020 Sabbatical. I really enjoyed your puzzles.
What you describe is the case of dedicating some razors and blades per slot, e.g. you use one mild and one aggressive razor per shave, paired with a sharp and a less sharp blade, respectively. Similar to that but with 5 categories would be to do a first pass with a straight and a second one with a DE.
Another possibility is to combine razors and blades more randomly. Say you have 7 razors, 3 brushes, 7 soaps, 4 blade types and you choose 2 razors and 2 blades. Then you get,
C(7,2) * 3 * 7 * C(4,2) = 21 * 3 * 7 * 6 = 2,646
where C(x,a) = x! / (r! * (x-r)!) denotes the possible combinations of picking r number of items out of a pool of x.